Suppose that f(x) is continuous on [2,4] and f'(x) exists on [2,4] and f'(C)= f(4)-f(2)/2, then C is in (2,4). T or F? Justify your answer

Question

Suppose that  f(x) is continuous on [2,4] and f'(x) exists on [2,4] and f'(C)= f(4)-f(2)/2, then C is in (2,4). T or F? Justify your answer

anonymous · Answer

have you heard of the mean-value theorem?

anonymous · Answer

yes,

anonymous · Answer

well, this is exactly the MVT (mean-value theorem), isn't it?

anonymous · Answer

yes, but im not sure how to justify the answer

anonymous · Answer

the MVT is justification

anonymous · Answer

ok thanks,, quick question f(X) = x/2-x derivation is f'(X)= -1x(2-x)^-2 (-1)

anonymous · Answer

is it$$f(x)=\frac{ x }{ 2-x }$$

anonymous · Answer

yeah

anonymous · Answer

so use the quotient rule, yeah?

anonymous · Answer

no but using the rule i got this f'(X)= -x(2-x)^-2 + (2-x)^-1

anonymous · Answer

$$f'(x)=\frac{ 1\cdot \left( 2-x \right)-x \cdot \left( -1 \right) }{\left( 2-x \right)^2 }=\frac{  2-x +x }{\left( 2-x \right)^2}=\frac{  2 }{\left( 2-x \right)^2}$$

anonymous · Answer

got it

anonymous · Answer

thanks

anonymous · Answer

you're welcome